Injection of Inner Oort Cloud Objects Into the Distant Kuiper Belt by Planet Nine

Draft version April 14, 2021 Typeset using LATEX twocolumn style in AASTeX62

Konstantin Batygin1 and Michael E. Brown1

1Division of Geological and Planetary Sciences California Institute of Technology, Pasadena, CA 91125, USA

ABSTRACT The outer solar system exhibits an anomalous pattern of orbital clustering, characterized by an approximate alignment of the apsidal lines and angular momentum vectors of distant, long-term stable Kuiper belt objects. One explanation for this dynamical conﬁnement is the existence of a yet- undetected planetary-mass object, “Planet Nine (P9)”. Previous work has shown that trans-Neptunian objects, which originate within the scattered disk population of the Kuiper belt, can be corralled into orbital alignment by Planet Nine’s gravity over ∼Gyr timescales, and characteristic P9 parameters have been derived by matching the properties of a synthetic Kuiper belt generated within numerical simulations to the available observational data. In this work, we show that an additional dynamical process is in play within the framework of the Planet Nine hypothesis, and demonstrate that P9-induced dynamical evolution facilitates orbital variations within the otherwise dynamically frozen inner Oort cloud. As a result of this evolution, inner Oort cloud bodies can acquire orbits characteristic of the distant scattered disk, implying that if Planet Nine exists, the observed census of long-period trans-Neptunian objects is comprised of a mixture of Oort cloud and Kuiper belt objects. Our simulations further show that although inward-injected inner Oort cloud objects exhibit P9-driven orbital conﬁnement, the degree of clustering is weaker than that of objects originating within the Kuiper belt. Cumulatively, our results suggest that a more eccentric Planet Nine is likely necessary to explain the data than previously thought.

Keywords: planets and satellites: dynamical evolution and stability

1 1. INTRODUCTION long-period (P & 4,000 yr) orbits in physical space Over the course of the past three decades, the dynam- (Batygin & Brown 2016). That is, trans-Neptunian ob- ical architecture of the solar system’s trans-Neptunian jects with semi-major axes in excess of 250 AU and in- region has steadily come into sharper focus. In turn, clinations smaller than 40 deg have apsidal lines and the unprecedented level of detail unveiled by obser- angular momentum vectors that cluster together in an vational surveys has repeatedly challenged our under- unexpected manner (Figure1). Although this clustering standing of the solar system’s markup, as well as the is visible by eye, it is important to note the degree of hitherto standard model of its long-term evolution (Sail- orbital confinement is not uniform, and instead depends lenfest 2020). The resulting paradigm shift towards a sensitively on dynamical stability. In particular, ob- instability-driven model of the solar system’s early dy- jects that experience comparatively rapid orbital diffu- namics has resolved much of the tension between theory sion due to periodic interactions with Neptune (shown as and observations, and today, the physical mechanisms green ellipses on Figure1) are much less tightly confined arXiv:2104.05799v1 [astro-ph.EP] 12 Apr 2021 that shaped the structure of the proximate (a . 50 AU) Kuiper belt have been broadly delineated (Morbidelli & 1 Simultaneous alignment of orbital planes and eccentricity vec- Nesvorný 2020 and the references therein). In contrast, tors requires the orbits to have similar longitudes of perihelion, a full understanding of the anomalous orbital architec- $, inclinations, i, and longitudes of ascending node, Ω. Impor- tantly, we note that the grouping of the arguments of perihelion, ture exhibited by distant trans-Neptunian objects – and ω = $ − Ω, was first pointed out by Trujillo & Sheppard(2014). its dynamical origins – remains elusive. Perhaps the most striking peculiarity displayed by the extended Kuiper belt is the approximate alignment of 2 than their stable and metastable counterparts2 (shown matches the observations. A subtle limitation of these as purple and gray ellipses on Figure1, respectively). calculations, however, is that they treat the solar sys- A number of explanations have been proposed for the tem as an isolated object, ignoring the effects of passing origins of the alignment depicted in Figure (1). Ar- stars, both during the solar system’s infancy and during guably the most rudimentary of them is the notion that its long-term field evolution, as well as the gravitational the alignment is not real, and is instead a figment of ob- tides of the birth cluster and the Galaxy. As such, all servational bias or statistical chance. To this end, indi- of the material that comprises the distant (a & 250 AU) vidual surveys – which have only searched limited areas Kuiper belt within the framework of these numerical ex- of the sky – have been unable to overcome their own periments is envisioned to have been sourced from inside observational biases sufficiently to rigorously determine of 30AU, and emplaced into the trans-Neptunian region the absence or presence of orbital alignment (Shankman by scattering off of the giant planets during the solar et al. 2017; Bernardinelli et al. 2020; Napier et al. 2021). system’s early transient instability (Levison et al. 2008; Nevertheless, a combined observability analysis of all Nesvorný 2018). available data shows that, after taking observational bi- Although the aforementioned picture reasonably rep- ases into account, distant KBOs are jointly clustered resents the evolution of objects with semi-major axes on in eccentricity and angular momentum vectors at the the order of a few hundred AU, more recent detections ∼ 99.8% significance level (Brown 2017; Brown & Baty- of trans-Neptunian objects (Sheppard & Trujillo 2016; gin 2019). Thus, if we interpret the picture insinuated Sheppard et al. 2019) increasingly point to a pronounced by the data at face value, the observed clustering re- abundance of long-period TNOs with a & 1000 AU. Cru- quires a sustained gravitational torque that maintains cially, this orbital domain borders the inner Oort cloud orbital alignment against differential precession induced (IOC) – a population of debris captured onto detached, by Jupiter, Saturn, Uranus and Neptune. long-term stable orbits during the solar system’s resi- In principle, such a torque could come from any dis- dence within its birth cluster (Kaib et al. 2011; Clement tant massive object, including a lopsided massive disk & Kaib 2020). The marked existence of such extremely of icy material (Madigan et al. 2018; Sefilian & Touma long-period members of the Kuiper belt demands that 2019), a primordial black hole (Scholtz & Unwin 2020), we consider the possibility that a fraction of the ob- an asymmetric dark matter halo, etc. In this work, we jects that comprise the population shown in Figure (1) opt for what we consider a somewhat less exotic alter- have been injected into the distant solar system from the native, and envision that the alignment of distant orbits outside, through an intricate interplay between Galactic is sustained by a yet-unseen planet, “Planet Nine”, oc- effects and Planet Nine’s gravity. cupying a mildly eccentric, slightly inclined orbit with a In this work, we investigate this hypothesis through period of order ∼ 10,000 years (Batygin & Brown 2016; numerical experimentation, with an eye towards under- Brown & Batygin 2016). We note however, that because standing how the process of mixing between the outer the dominant mode of P9-induced dynamics is secular Kuiper belt and the IOC alters the degree of orbital con- (i.e., stemming from phase-averaged gravitational po- finement facilitated by Planet Nine. In particular, we tential of P9), our results are not sensitive to the com- simulate the formation of the inner Oort cloud within position of P9 – only its mass and its orbit. the solar system’s birth cluster (section3) and subject the resulting population of debris to P9-forced evolu- 2. HYPOTHESIS tion (section4). As we describe below, our results in- In a recent effort (Batygin et al. 2019), we have car- dicate that although material injected into the distant ried out an extensive suite of N−body simulations of Kuiper belt by Planet Nine exhibits orbital clustering, the solar system’s long-term evolution in presence of the degree of confinement is inferior compared to that Planet Nine, and have demonstrated that a synthetic exhibited by objects that originate within the scattered disk. These findings introduce an added degree of un- Kuiper belt sculpted by a m9 = 5 M⊕ P9 residing on a certainty into our attempts to accurately estimate the a9 = 500 AU, e9 = 0.25, i9 = 20 deg orbit adequately physical and orbital properties of Planet Nine.

2 For the purposes of this work, we deﬁne the tiers of orbital stability as follows: if more than 20% of the clones of a given 3. FORMATION OF THE INNER OORT CLOUD object eject over the course of a 4 Gyr integration, an object is Statistically speaking, the vast majority of stars form deemed unstable. If more than 20% of the clones of a given object change their semi-major axes by a factor of two during a 4 Gyr in stellar associations (Lada & Lada 2003; Porras et al. integration but do not eject, an object is deemed metastable. If 2003). The solar system is no exception to this rule, neither of these events occurs, we deem such an object stable. and the enrichment of meteorites in short-lived radio- 3

Figure 1. Census of trans-Neptunian objects with a > 250 AU, q > 30 AU, and i < 40 deg. The orbits of objects that experience rapid orbital diffusion are shown in green, while the orbits of stable and slowly diffusing KBOs are depicted in purple and gray, respectively. The dotted orbits correspond to KBOs whose parameters and uncertainties are not reported in the JPL Small Body Database, and whose stability was estimated from a single 4 Gyr integration of the nominal orbit rather than similar integrations of ten clones of each object. The i − Ω polar inset informs the tilt of the orbital angular momentum vectors. Stable and metastable objects have apsidal lines that cluster around h$i ∼ 60 deg, and form a mean plane that is inclined with respect to the ecliptic by hii ∼ 10 deg, with a mean longitude of ascending node of hΩi ∼ 90 deg. genic isotopes is among the most direct lines of evidence potential exert significant perturbations on solar system that point to the notion that the sun’s birth environ- objects. Correspondingly, any planetesimals that would ment likely played an important role in shaping the so- have been scattered out to large heliocentric distances by lar system’s large-scale structure (see Adams 2010 for a the forming giant planets would have experienced some review). In fact, if the process of planetary accretion un- degree of modulation by cluster effects (Morbidelli & folded while the sun was still embedded within its birth Levison 2004). In particular, published numerical sim- cluster, the formation of a perihelion-detached cloud of ulations show that as long as the solar system contin- debris would have been all but inevitable. ues to reside within its birth cluster, the population of An inescapable consequence of the solar system’s resi- planetesimals that attain r & 1000 AU through outward dence within a cluster of stars is that at sufficiently large scattering remain entrained in a quasi-steady state cy- orbital radii, passing stars and the cluster’s mean-field cle of enrichment through giant planet scattering, cir- 4

Figure 2. Sequence of events modeled within this work. A perihelion-detached population of trans-Neptunian objects forms while the sun is embedded within its birth cluster. Subsequently, over the multi-Gyr lifetime of the solar system, Planet Nine slowly diminishes the perihelia of a subset of these extremely long-period objects, mixing them into the observed census of Kuiper belt objects. Orbits shown on the left and middle panels of the Figure are taken from the t = 4 Myr model of the inner Oort cloud (IOC) shown in Figure (3). Orbits depicted on the right panel of the Figure correspond to the same sample of orbits evolved for t > 2 Gyr, and shown as dynamical footprints Figure (2). cularization through cluster effects, and stripping by gin et al. 2020 and the references therein). We further passing stars (Brasser et al. 2006, 2012). As soon as assumed the sun to steadily orbit the√ cluster core at the 1/3 the sun leaves its birth environment, however, this pro- half-mass radius of r = (1 + 2 )/ 3 c ≈ 0.33 pc on a cess ceases to operate, terminating the generation of the circular orbit, and ignored the long-term dissipation of IOC. the system, thereby holding the stellar number density 3 We have simulated the sequence of events depicted in fixed at n ≈ 1400/pc . Finally, we adopted a velocity Figure (2), following the procedure outlined in Brasser dispersion of hvi = 1 km/s for all 19 stellar species4 that et al.(2006). Our simulations included Jupiter and Sat- comprised the model stellar mass function of Heisler et urn, residing at aJ = 5.5 AU and aS = 9 AU respectively, al.(1987), and intermittently injected passing stars into as well as a sea of N = 105 massless planetesimals, span- the simulation domain following the method outlined in ning the 4.5−12 AU range in heliocentric distance. Both Heisler et al.(1987). Unlike these authors, however, we of the planets as well as the test particles were initial- did not employ the impulse approximation, and instead ized on circular and coplanar orbits, and were integrated resolved stellar flybys through self-consistent N-body in- using the conservative variant of the Bulisch-Stoer algo- tegration (adopting the simulation parameters described rithm implemented in the mercury6 gravitational dy- above), setting the stellar injection/ejection radius to namics software package (Chambers 1999). The initial 0.1 pc. Cumulatively, these choices yield a characteris- 2 time step and integration accuracy parameter were set tic stellar encounter rate of Γ ∼ n π b? hvi ∼ 50/Myr. to ∆t = 102 days and = 10−12 respectively. For completeness, we note a number of additional sim- In addition to planetary perturbations, we accounted plifying assumptions that were made in our calculations. for the effects of passing stars with impact parameters First, we assumed an essentially perfect star-formation smaller than b? 6 0.1 pc, as well as the mean-field po- efficiency, such that that the total mass of cluster is tential of the solar birth cluster. For definitiveness, we equal to the mass comprised by the stars. In practice, modeled the cluster as a Plummer sphere, setting the total mass to M∞ = 1200 M and Plummer radius to – values comparable to the properties of the Trapezium cluster, c = 0.35 pc in an effort to approximately mimic the embedded within the ONC. Moreover, with these parameters, the 3 properties of the Orion Nebular Cluster (see also Baty- mean stellar number density interior to the M/M∞ = 95% radius evaluates to hni ≈ 100/pc3, in agreement with observations (Hillenbrand & Hartmann 1998). 3 Note that these choices correspond to a cluster core radius of 4 White dwarfs were neglected from the stellar mass function. 4 3 rc ≈ 0.23 pc and central number density of nc ≈ 1.7 × 10 /pc 5

Figure 3. A representative inner Oort cloud created in our simulations. Multi-colored points depict perihelion-detached population of TNOs, generated within ten distinct realizations of the model cluster. Crudely speaking, cluster effects facilitate perihelion lifting, excitation of inclination, and clustering of the argument of perihelion for a & 500 AU objects. As the sun’s cluster residence time increases, both the inner and outer edges of the detached population inch towards smaller heliocentric distances. this correspondence is of little dynamical consequence, efficient perihelion-detachment mechanism for outward since Kozai-Lidov type variations in particle parameters scattered planetesimals. due to the cluster’s mean-field potential (see e.g., sec- To smooth over statistical fluctuations inherent to tion 2 of Batygin et al. 2020) represent only a small, each recapitulation of stochastic evolution facilitated by longer-term correction to direct perturbations exerted stellar encounters, we split the test particle population by the stars, meaning that modulating the mass bud- of our model into ten separate disks (each composed of get of the gas is unlikely to appreciably change the N = 104 particles), and integrated each system within simulations outcomes. Second, we approximated the a distinct realization of the cluster. Each integration IMF within the cluster as being similar to the present- was carried out over a timespan of τ = 10 Myr. This day mass function with the exception of omitting white timescale is simultaneously comparable to the charac- dwarfs. While a more thorough treatment of the IMF teristic lifetime of a typical protoplanetary disk (Haisch is possible, this assumption is unlikely to manifest as et al. 2001), and close to the upper bound on the sun’s the dominant source of error in our calculations. Fi- permissible residence time within the model cluster. Im- nally, we ignored the potential presence of P9 itself – portantly, the latter constraint comes from a dynami- and its accompanying perturbations – during the clus- cal limit on the product of stellar number density and 4 3 ter stage. This simplification can be justified through a cluster residence time of hni τ . 2 × 10 Myr/pc , de- timescale argument: published simulations indicate that rived from the inclination dispersion of the cold clas- P9-induced evolution unfolds on ∼Gyr timescales – far sical Kuiper belt (Batygin et al. 2020), and is largely longer than lifetimes of a typical birth cluster. This independent of the processes modeled here. With the means that even if Planet Nine were extant during the adopted parameters, we found that a substantial IOC sun’s cluster phase, its gravity would not constitute an forms within ∼ 1 Myr, and reaches steady-state in terms 6

Figure 4. Synthetic long-period Kuiper belt. The top panels show the t > 2 Gyr, 40 AU < q < 100 AU, i < 40 deg dynamical footprints of particles generated solely by diffusion of inner Oort cloud objects to smaller orbital radii. Each set of trajectories is color coded by the cluster residence time of the inner Oort cloud model from which they were generated. The middle panels depict the smoothed density histogram produced by combining all of the results. The depicted data points correspond to the q > 40 AU subset of orbits illustrated in Figure (1), and are shown using the same color scheme. Notably, with the exception of a single object (2015 KG163), the depicted data correspond to TNOs that do not experience rapid orbital diffusion induced by Neptune. The bottom panels show equivalent orbital footprints from a control simulation employing the same Planet Nine parameters, but with all particles initialized uniformly within the a ∈ (100, 800) AU range. Cumulatively, our results indicate that although P9-facilitated apsidal alignement and angular momentum vector clustering is clearly evident in the IOC-fed simulation results, it is considerably weaker than orbital confinement exhibited by particles sourced from the scattered disk population of Kuiper belt. 7 of particle count at ∼ 2 Myr. More specifically, between jects steadily march towards lower heliocentric dis- 2 and 10 Myr, the number of particles with semi-major tances in time, crudely tracking the ever-decreasing axes in excess of 40 AU and perihelion distance greater impact parameter of the closest stellar encounter p than 40 AU fluctuates between 200 and 300, although it bmin ∼ 1/ π n hvi t. Overall, we find that the dynam- is important to understand that the particles that com- ical decoherence timescale of the detached population is prise the cloud are continuously in flux. somewhat shorter than a million years, and adopt snap- A representative snapshot of the dynamical state of shots of the distant solar system at 1, 2, 3, ..., 10 Myr as the IOC at t = 4 My is shown in Figure (3). The pseudo-independent models of the IOC. top left panel depicts particles eccentricities as a function of semi-major axes. Constant-perihelion curves 4. P9-INDUCED EVOLUTION corresponding to q = 5, 10, 20, 30, and 40 AU are also In absence of Planet Nine, the cloud of distant debris outlined, and the outward transport of particles along created by the interplay between giant planet scatter- the Saturnian scattered disk is evident. Objects with ing and cluster perturbations would essentially remain q 40 AU are shown as white circles, while perturbed 6 frozen over the main-sequence lifetime of the sun. Grav- particles with q > 40 AU are denoted with filled points, itational perturbations due to P9, however, can steadily each color-coded in accord with the cluster realization diminish the perihelia of such detached objects, bring- within which it was generated. A smoothed orange den- ing them into the fold of the scattered disk population of sity histogram underlying the perihelion-detached pop- the Kuiper belt. Here, we have simulated this process, ulation of particles is also illustrated in each panel. Par- self-consistently accounting for the N−body dynamics ticle inclinations are depicted on the top right panel of driven by Neptune, Planet Nine, passing stars within Figure (3) and show that although stellar encounters the Galactic field (once again, following the methodol- can generate retrograde objects, the kernel of the IOC ogy of Heisler et al. 1987), as well as the effect of the is comprised of objects with inclinations on the order of Galactic tide (implemented as in Nesvornýet al. 2017; a few tens of degrees. see also Heisler & Tremaine 1986; Wiegert & Tremaine The argument of perihelion and longitude of ascend- 1999; Clement & Kaib 2020). As noted in the afore- ing node are portrayed in the bottom left and bottom mentioned references, this effect is most sensitive to the right panels respectively. While the nodal structure of vertical component of the acceleration, which is largely the cloud is fully randomized, the argument of peri- determined by the mean local Milky Way disk density, helion exhibits distinct clustering around ω = 0 and which we set to ρ = 0.1 M /pc3. ω = 180 deg. Unlike the dynamical grouping shown in MW For definitiveness, we adopted the best-fit P9 param- Figure (1), these patterns are readily understood. Recall eters from (Batygin et al. 2019): m = 5 M , a = that within the context of our simulations, particles at- 9 ⊕ 9 500 AU, e = 0.25 and i = 20 deg. Orbit-averaged tain large heliocentric distances exclusively by scattering 9 9 effects of Jupiter, Saturn, and Uranus were also in- off of giant planets, meaning that by the time they get cluded in the calculations as an effective J moment perturbed by passing stars, their orbits are nearly radial 2 of the sun. As discussed in Batygin et al.(2019), this but planar, with e ∼ 1−q /a ∼ 0.99 and i ∼ 0. Because S numerical scheme constitutes an adequate compromise Ω is ill-defined at i = 0, even a weak stellar flyby can between computational cost and accuracy, allowing us drastically alter the longitude of ascending node. Signif- to increase the initial tilmestep in our simulations to icantly changing ω on the other hand, would require a ∆t = 3000 days. perturbation to rotate the nearly-radial orbit away from We integrated each of the ten variants of the IOC dis- the orbital plane of Jupiter and Saturn by a large angle. cussed in the previous section for 4.5 Gyr, and found Instead, Kepler’s second law insinuates that interactions that over the lifetime of the sun, a significant frac- are most likely to occur at aphelion, where an impul- tion (that is, on the order of 20%) of the IOC gets in- sive change in particle velocity can easily modulate the jected into the distant (a > 250 AU) Kuiper belt, with semi-major axis and eccentricity, but not alter the or- approximately 10% injected in the final 2 Gyr. Dy- bital orientation. As a result, even after being perturbed namical footprints with t > 2 Gyr of injected parti- by passing stars, particles continue to come to aphelion cles that satisfy crude stability and observability criteria close to the orbital plane of the giant planets, thereby (40 AU < q < 100 AU, i < 40 deg) are shown in the top maintaining ω = 0 or ω = 180 deg. panels of Figure (4), where the color of the points in- The dynamical state of the IOC appears similar at forms the IOC model adopted for the initial conditions. other times in the integration, with the caveat that More specifically, the panels on the left hand side depict the inner and outer boundaries of the perturbed ob- the particles’ longitude of perihelion, relative to that of 8

Figure 5. Apsidal clustering and semi-major axis distributions of simulated particles satisfying a > 250 AU, 40 AU < q < 100 AU, i < 40 deg and t > 2 Gyr. The multi-colored curves show the ∆$ and a probability density functions of inward-injected Oort cloud objects obtained in our simulations. The black curves depict the probability density functions associated with synthetic KBOs that are initialized within the scattered disk, and evolved with identical (m9 = 5 M⊕, a9 = 500 AU, e9 = 0.25, i9 = 20 deg) P9 parameters. Although the dynamical forcing is the same, objects that originate within the Kuiper belt exhibit significantly tighter apsidal clustering in our simulations than do injected IOC objects. Injected IOC objects further display a somewhat more extended semi-major axis distribution than our control simulation. Planet Nine, while the polar plots on the right inform gitude of perihelion exhibiting visible clustering around the angular momentum vectors of particle orbits. ∆$ ∼ 180 deg (Figure4). Similarly, the inclination - Overall, IOC models that correspond to solar clus- node trajectories of these objects circulate around the ter residence time of t > 2 Myr yield consistent re- P9-forced equilibrium in phase space, which can man- sults, although the relevant particle count in each case ifest in an apparent clustering of the longitudes of as- is relatively small (Table1). Thus, in order to more cending node when viewed from the ecliptic. At the clearly portray the orbital distribution of the long-period same time, it is important to note that the patterns of Kuiper belt generated by inward-injection of distant de- orbital clustering shown in Figure (4) are considerably bris, we show a smooth density histogram of our com- less pronounced than those created in P9 simulations bined5 simulation data in the middle panels of Figure where distant KBOs are sourced entirely from the scat- (4). The census of known long-period KBOs satisfy- tered disk. ing the same perihelion and inclination cuts is over- To be more specific, the fraction of dynamical foot- plotted on the digram, using the same color scheme prints that fall within ±90 deg of exact apsidal anti- as that in Figure (1). Finally, the bottom panels of alignment with respect to P9 in the middle panel of Figure (4) depict the results of a control simulation us- Figure (4) is a mere f$ = 67%. By comparison, our ing the same P9 parameters, but with particle initial control simulation with initial semi-major axes of par- conditions drawn uniformly from the q ∈ (30, 100) AU ticles drawn from the a ∈ (100AU, 800AU) range yields and a ∈ (100, 800) AU ranges, and an initial inclination f$ = 88%. The confinement of angular momentum dispersion corresponding to a half-gaussian distribution vectors in our IOC-fed simulations is similarly weak: with a standard deviation of 15 deg. the mean inclination and the associated rms disper- Cumulatively, our simulations elicit two key results. sion exhibited by injected particles are hii ≈ 5 deg and First, the injected particles clearly follow typical P9- irms ≈ 26 deg, respectively. Meanwhile, our scattered driven dynamical evolution (see Batygin & Brown 2016; disk-fed control simulation shows a more tightly clus- Batygin & Morbidelli 2017; Khain et al. 2018, 2020; tered particle distribution, characterized by hii ≈ 9 deg Li et al. 2018; Hadden et al. 2018) with relative lon- and irms ≈ 19 deg. Thus, taken at face value, our calculations suggest that the presence of a significant IOC could partially obfuscate the dynamical signal generated 5 Although the distant Kuiper belt sourced from the t = 1 Myr model of the IOC shows virtually no apsidally anti-aligned objects by Planet Nine. with q < 100 AU, the particle count is a factor of a few smaller The difference in the statistical measures of orbital than the other runs, meaning that its contribution is insignificant. clustering between inward-injected IOC objects and 9

Table 1. A summary of orbital conﬁnement metrics for each IOC-fed simulation. The percentages in parentheses indicate the fraction of down-sampled control simulations that yield metrics as low or lower than the reported numerical experiments.

IOC model 1 2 3 4 5 N † 11 15 15 26 37

f$ 0.46 (0%) 0.62 (0%) 0.69 (0%) 0.64 (0%) 0.75 (0%) hii (deg) 20.2 (100%) 9.2 (40%) 10.1 (67%) 4.6. (0%) 4.2 (0%)

hii/irms 0.98 (100%) 0.37 (23%) 0.43 (26%) 0.20 (0%) 0.16 (0%) IOC model 6 7 8 9 10 N † 31 28 30 30 37

f$ 0.70 (0%) 0.65 (0%) 0.71 (0%) 0.64 (0%) 0.67 (0%) hii (deg) 7.4 (10%) 0.5 (0%) 3.8 (0%) 10.7 (97%) 3.6 (0%)

hii/irms 0.29 (0%) 0.02 (0%) 0.14 (0%) 0.45 (47%) 0.13 (0%) outward-scattered Kuiper belt objects within the con- cases consistent with our control simulation, apsidal text of P9 simulations begs the question of whether or clustering in these simulations is consistently inferior. not this discrepancy can be attributed to a difference To illustrate this point further, we show the ∆$ distri- in particle count among the various numerical experi- butions of simulated particles in the left panel of Fig- ments. A cursory examination of our simulation suite, ure (5), where the control and IOC-fed simulations are however, suggests that this is unlikely to be the case. shown with black and multi-colored curves, respectively. Out of the 2440 objects initialized within the full set of A second important result of our numerical experi- our IOC-fed simulations, a total of 260 particles attain ments is that steady infusion of IOC material into the orbits that satisfy the aforementioned observability cri- Kuiper belt can enhance the prevalence of scattered disk teria at some time in excess of t > 2 Gyr. The same is objects with semi-major axes in excess of a & 1, 000 AU. true for 227 (out of 1000 initial) particles in our con- That is, although the semi-major axis distribution of trol simulation, implying that the dynamical footprints dynamical footprints shown in Figure (4) peaks in the depicted in the top and bottom panels of Figure (4) a ∼ 250 − 500 AU range, the tail of the distribution ex- are derived from a comparable number of independent tends beyond a & 3000 AU, as shown in the right panel tracers. of Figure (5). Relative to the results of our control sim- In order to quantify the statistical significance of these ulation, these IOC-fed calculations can readily yield a differences further, we computed the three measures of factor of ∼ 5 enhancement in the semi-major axis distri- dynamical confinement – f$, hii, hii/irms – for each bution of distant KBOs at a = 2000 AU and more than IOC-fed model independently, and computed the like- an order of magnitude increase at a = 3000 AU. Thus, lihood that values as low as these can arise within the an extended dispersion of KBO semi-major axes may context for an appropriately re-sampled control simula- indirectly hint at the operation of P9-facilitated modu- tion, where the particles are taken to originate within lation of the IOC. the Kuiper belt. In particular, we down-sampled the Simultaneously, however, it must be understood that control simulation such that the number of indepen- this determination is preliminary. Recall that while the dent particles that satisfy our observability criteria at initial conditions of our IOC-fed calculations are gener- t > 2 Gyr, N †, are the same as that in each IOC-fed ated through self-consistent numerical experiments, the simulation, and computed the orbital confinement met- initial conditions of our control simulation were cho- rics thirty times, randomly reshuffling the particles. We sen in an ad-hoc manner. Because the starting semi- then recorded the fraction of such reshuffled simulations major axis distribution of our control calculation is trun- that yield values of f$, hii, and hii/irms equal to or lower cated at 800 AU, the results may be underestimating the than those obtained within the corresponding IOC-fed breadth of the true semi-major axis distribution that can calculation. be generated through P9 induced evolution of the scat- These results are summarized in Table1. As can be tered disk. On the other hand, if the intrinsic surface deduced from these calculations, it is improbable that density of the scattered disk diminishes more steeply the differences in the synthetic Kuiper belts sourced than ΣSD ∝ 1/r (as is suggested for example by the from inward injection vs. outward scattering are spu- results of Nesvornýet al. 2017), then our control sim- rious. In particular, although the inclination-node clus- ulation is likely to be overestimating the extent of the tering generated in IOC-fed calculations are in some high-a tail of the orbital distribution. A distinct degen- 10 eracy relevant to the surface density profile of distant bital confinement. However, the degree of clustering is icy bodies arises from the possibility that the presence considerably weaker. of the classical Oort cloud may generate an uptick in This discrepancy carries important observational con- TNOs with a & 1,000 AU that is completely indepen- sequences. Because the extent of apsidal clustering of dent of P9-facilitated evolution. Therefore, more precise distant objects increases with e9 (Batygin et al. 2019), calculations that account for the existence of the clas- contamination of the distant Kuiper belt by poorly- sical Oort cloud and employ a self-consistent estimate confined IOC objects implies that a more eccentric and of the initial conditions of the scattered disk, are neces- distant Planet Nine may be required to explain the sary to confidently elucidate the difference between the data, than is insinuated by the existing literature. How- semi-major axis distributions formed by IOC-fed and ever, the magnitude to which a best-fit P9 orbit must scattered disk-sourced populations of distant TNOs. be modified depends on a yet-unconstrained fraction of KBOs that are sourced from the IOC. Irrespective of this notion, it is important to note that Planet Nine’s eccentricity cannot be enhanced by a large margin without vi- 5. CONCLUSION olating other constraints, including the over-production In this work, we have carried out a series of numeri- of highly inclined scattering objects that ensues for suf- cal experiments in an effort to quantify the injection of ficiently high values of e9, as pointed out by Kaib et inner Oort cloud material into the distant Kuiper belt al.(2019). Cumulatively, the proof-of-principle results via P9-induced dynamical evolution. To this end, we reported herein introduce an important, additional de- have constructed a model IOC, accounting for the grav- gree of uncertainty in our efforts to pin-down the orbit itational effects of the sun’s birth cluster (section3), of Planet Nine. and have followed the long-term evolution of this population of distant debris, subject to perturbations from We are thankful to Alessandro Morbidelli, Dar- the giant planets, passing stars, galactic tide, as well ryl Seligman, Juliette Becker, Fred Adams, David as P9 (section4). The results of our simulations are Nesvorný,and Eduardo Marturet for insightful discus- readily summarized: if Planet Nine exists, then it fa- sions. We are additionally grateful to David Nesvorný cilitates steady variations in the orbits of a & 1000 AU for sharing his implementation of Galactic tide acceler- IOC bodies, bringing a fraction of them into the fold of ations. We thank the anonymous referee for providing a the observable component of the distant scattered disk. thorough and insightful report. K.B. is grateful to Cal- Much like the conventional members of the long-period tech, the David and Lucile Packard Foundation, and the Kuiper belt, these injected objects exhibit P9-driven or- Alfred P. Sloan Foundation for their generous support.

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